\begin{equation} \label{eq:sum_of_sine_waves}
    H(\mathbf{p},t) = \sum_i{
        A_i \sin \, (
            (\mathbf{p} \cdot \mathbf{d}_i) w_i
            + t \varphi_i
        )
    } , \mathbf{p} = 
\end{equation}

\begin{equation} \label{eq:sum_of_sine_waves_position}
    \mathbf{H}(\mathbf{p},t) = \begin{bmatrix}
        p_x&
        H(\mathbf{p},t)&
        p_z
    \end{bmatrix}
\end{equation}

\begin{equation} \label{eq:sum_of_sine_waves_normal}
    \mathbf{\hat{H}}(\mathbf{p},t) = \begin{bmatrix}
        -\frac{\partial}{\partial p_x} H(\mathbf{p},t)&
        1&
        -\frac{\partial}{\partial p_z} H(\mathbf{p},t)
    \end{bmatrix}
\end{equation}

\begin{equation} \label{eq:gerstner_wave}
    G_j(\mathbf{p},t) = \sum_i{
        d_{i,j} Q_iA_i \cos \, (
            (\mathbf{p} \cdot \mathbf{d}_i) w_i
            + t \varphi_i
        )
    }
\end{equation}

\begin{equation} \label{eq:gerstner_wave_position}
    \mathbf{G}(\mathbf{p},t) = \begin{bmatrix}
        p_x + G_x(\mathbf{p},t)\\
        H(\mathbf{p},t)\\
        p_z + G_z(\mathbf{p},t)
    \end{bmatrix}
\end{equation}

\begin{equation} \label{eq:gerstner_wave_normal}
    \mathbf{\hat{G}}(\mathbf{p},t) = \begin{bmatrix}
        -\sum_i{d_{i,x} A_iw_i \cos \, (\mathbf{G}(\mathbf{p},t) \cdot \mathbf{d}_i)}\\
        1 - \sum_i{Q_iA_iw_i \sin \, (\mathbf{G}(\mathbf{p},t) \cdot \mathbf{d}_i)}\\
        -\sum_i{d_{i,z} A_iw_i \cos \, (\mathbf{G}(\mathbf{p},t) \cdot \mathbf{d}_i)}
    \end{bmatrix}
\end{equation}
